Skip to main content
SpringerLink
Log in

A Maximum Principle Applied to Quasi-Geostrophic Equations

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the initial value problem for dissipative 2D Quasi-geostrophic equations proving local existence, global results for small initial data in the super-critical case, decay of Lp-norms and asymptotic behavior of viscosity solution in the critical case. Our proofs are based on a maximum principle valid for more general flows.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baroud, Ch. N., Plapp, B.B., She, Z.-S., Swinney, H.L.: Anomalous self-similarity in a turbulent rapidly rotating fluid. Phys. Rev. Lett. 88, 114501 (2002)

    Article  Google Scholar 

  2. Berselli, L.: Vanishing viscosity limit and long-time behavior for 2D Quasi-geostrophic equations. Indiana Univ. Math. J. 51 (4), 905–930 (2002)

    MATH  Google Scholar 

  3. Chae, D.: The quasi-geostrophic equation in the Triebel-Lizorkin spaces. Nonlinearity 16 (2), 479–495 (2003)

    Article  MATH  Google Scholar 

  4. Chae, D., Lee, J.: Global Well-Posedness in the super critical dissipative Quasi-geostrophic equations. Commun. Math. Phys. 233, 297–311 (2003)

    MATH  Google Scholar 

  5. Coifman, R., Meyer, Y.: Au delà des operateurs pseudo-differentiels. Asterisqué 57, Paris: Société Mathmatique de France, 1978, pp. 154

  6. Coifman, R., Meyer, Y.: Ondelettes et operateurs. III. (French) [Wavelets and operators. III] Operateurs multilinaires. [Multilinear operators] Actualits Mathmatiques. [Current Mathematical Topics] Paris: Hermann, 1991

  7. Constantin, P.: Energy Spectrum of Quasi-geostrophic Turbulence. Phys. Rev. Lett. 89 (18), 1804501–4 (2002)

    Google Scholar 

  8. Constantin, P., Cordoba, D., Wu, J.: On the critical dissipative Quasi-geostrophic equation. Indiana Univ. Math. J. 50, 97–107 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Constantin, P., Wu, J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 30, 937–948 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cordoba, D.: Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. of Math. 148, 1135–1152 (1998)

    MathSciNet  MATH  Google Scholar 

  12. Cordoba, A., Cordoba, D.: A pointwise estimate for fractionary derivatives with applications to P.D.E. Proc. Natl. Acad. Sci. USA 100 (26), 15316–15317 (2003)

    Article  Google Scholar 

  13. Cordoba, D., Fefferman, C.: Growth of solutions for QG and 2D Euler equations. J. Am. Math. Soc. 15 (3), 665–670 (2002)

    Article  MATH  Google Scholar 

  14. Dinaburg, E.I., Posvyanskii, V.S., Sinai, Ya.G.: On some approximations of the Quasi-geostrophic equation. Preprint.

  15. Held, I., Pierrehumbert, R., Garner, S., Swanson, K.: Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20 (1995)

    MathSciNet  MATH  Google Scholar 

  16. Kato, T., Ponce, G.: Commutators estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988)

    MathSciNet  MATH  Google Scholar 

  17. Pedlosky, J.: Geophysical Fluid Dynamics. New York: Springer-Verlag, 1987

  18. Resnick, S.: Dynamical problems in nonlinear advective partial differential equations. Ph.D. thesis, University of Chicago, Chicago 1995

  19. Schonbek, M.E., Schonbek, T.P.: Asymptotic behavior to dissipative quasi-geostrophic flows. SIAM J. Math. Anal. 35 (2), 357–375 (2003)

    Article  MATH  Google Scholar 

  20. Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton NJ: Princeton University Press, 1970

  21. Stein, E., Zygmund, A.: Boundedness of translation invariant operators on Holder and Lp-spaces. Ann. of Math. 85, 337–349 (1967)

    MATH  Google Scholar 

  22. Wu, J.: Dissipative quasi-geostrophic equations with Lp data. Electronic J. Differ. Eq. 56, 1–13 (2001)

    Google Scholar 

  23. Wu, J.: The quasi-geostrophic equations and its two regularizations. Comm. Partial Differ. Eq. 27 (5–6), 1161–1181 (2002)

    Google Scholar 

  24. Wu, J.: Inviscid limits and regularity estimates for the solutions of the 2-D dissipative Quasi- geostrophic equations. Indiana Univ. Math. J. 46 (4), 1113–1124 (1997)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Antonio Córdoba

  2. Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas, Serrano 117, 28006, Madrid, Spain

    Diego Córdoba

Authors
  1. Antonio Córdoba
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Diego Córdoba
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by P. Constantin

Partially supported by BFM2002-02269 grant.

Partially supported by BFM2002-02042 grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Córdoba, A., Córdoba, D. A Maximum Principle Applied to Quasi-Geostrophic Equations. Commun. Math. Phys. 249, 511–528 (2004). https://doi.org/10.1007/s00220-004-1055-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-004-1055-1

Keywords

Search

Navigation